After apprehending the suspect Monday in a fatal weekend shooting of four people at a Nashville-area Waffle House, authorities are looking into whether his father broke federal laws by giving him back his weapons, including the semiautomatic rifle used in the attack, that were legally taken away last year.

“He should not have had weapons,” said Marcus Watson, special agent in charge of the Nashville Field Division of the U.S. Bureau of Alcohol, Tobacco, Firearms and Explosives, referring to Travis Reinking, the suspect now...